## Character table for point group D8d

 D8d E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8d linear functions,rotations quadraticfunctions cubicfunctions A1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 - A2 +1 +1 +1 +1 +1 +1 +1 +1 +1 -1 -1 Rz - - B1 +1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 - - - B2 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 z - z3, z(x2+y2) E1 +2 +2cos(1/8) +(2)½ +2cos(3/8) 0 -2cos(3/8) -(2)½ -2cos(1/8) -2 0 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)] E2 +2 +(2)½ 0 -(2)½ -2 -(2)½ 0 +(2)½ +2 0 0 - (x2-y2, xy) - E3 +2 +2cos(3/8) -(2)½ -2cos(1/8) 0 +2cos(1/8) +(2)½ -2cos(3/8) -2 0 0 - - [x(x2-3y2), y(3x2-y2)] E4 +2 0 -2 0 +2 0 -2 0 +2 0 0 - - - E5 +2 -2cos(3/8) -(2)½ +2cos(1/8) 0 -2cos(1/8) +(2)½ +2cos(3/8) -2 0 0 - - - E6 +2 -(2)½ 0 +(2)½ -2 +(2)½ 0 -(2)½ +2 0 0 - - [xyz, z(x2-y2)] E7 +2 -2cos(1/8) +(2)½ -2cos(3/8) 0 +2cos(3/8) -(2)½ +2cos(1/8) -2 0 0 (Rx, Ry) (xz, yz) -

 Cs , C2 (2) , C4 , C8 , D2 , D4 , D8 , C2v , C4v , C8v , S16 Number of symmetry elements h = 32 Number of irreducible representations n = 11 Abelian group no Number of subgroups 12 Number of distinct subgroups 11 Subgroups (Number of different orientations) Optical Isomerism (Chirality) no Polar no

## Reduction formula for point group D8d

Type of representation

general 3N vib

E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8d

## Multipoles

dipole (p) B2+E1 A1+E2+E7 B2+E1+E3+E6 A1+E2+E4+E5+E7 B2+E1+E3+E4+E5+E6 A1+E2+E3+E4+E5+E6+E7 B2+E1+E2+E3+E4+E5+E6+E7 A1+B1+B2+E1+E2+E3+E4+E5+E6+E7 A1+A2+B2+E1+E2+E3+E4+E5+E6+2E7